Suppose that the graph of a certain function, $y=f(x)$, has the property that if it is shifted $20$ units to the right, then the resulting graph is identical to the original graph of $y=f(x)$.

What is the smallest positive $a$ such that if the graph of $y=f\left(\frac x5\right)$ is shifted $a$ units to the right, then we know that the resulting graph is identical to the original graph of $y=f\left(\frac x5\right)$?
Explanation: The stated property of $f(x)$ can be written as an equation which holds for all $x$: $$f(x-20) = f(x).$$We are looking for the smallest positive $a$ such that the equation $$f\left(\frac{x-a}5\right) = f\left(\frac x5\right)$$holds for all $x$. Rewriting this equation as $$f\left(\frac x5-\frac a5\right) = f\left(\frac x5\right),$$we see that it is implied by the known property of $f(x)$ if $\frac a5$ is equal to $20$ (or a multiple of $20$), or in other words, if $a$ is equal to $100$ (or a multiple of $100$). So, the smallest positive $a$ for which we know that this property holds is $a=\boxed{100}$.